My hotel had a swimming pool, and I wanted to host a poolside party for my birthday. I wanted to fill the pool using three pipes. Two pipes (A and B) could fill the pool in 4 hours and 9 hours. Another pipe can empty the pool in 6 hours. By mistake I opened all three pipes at 3 pm and realised at 3.30 pm. So, I closed it.
When would the pool be completely filled with water?

Let's break down the situation to find out when the pool would be completely filled with water.

### Given:
- Pipe A can fill the pool in 4 hours.
- Pipe B can fill the pool in 9 hours.
- Pipe C can empty the pool in 6 hours.

You opened all three pipes at 3:00 PM and closed them at 3:30 PM. Afterward, only the filling pipes (A and B) remain open.

---

#### Step 1: Work done by each pipe in 1 hour
- Pipe A's rate of filling = \( \frac{1}{4} \) of the pool per hour.
- Pipe B's rate of filling = \( \frac{1}{9} \) of the pool per hour.
- Pipe C's rate of emptying = \( \frac{1}{6} \) of the pool per hour.

#### Step 2: Net rate of filling with all three pipes open
With all three pipes open, the net rate of filling is:
\[
\text{Net rate} = \left( \frac{1}{4} + \frac{1}{9} - \frac{1}{6} \right)
\]
We need to simplify this:

\[
\frac{1}{4} = 0.25, \quad \frac{1}{9} \approx 0.111, \quad \frac{1}{6} \approx 0.167
\]
\[
\text{Net rate} = 0.25 + 0.111 - 0.167 = 0.194 \text{ (approximately)}.
\]

So, the net rate of filling when all three pipes are open is approximately 0.194 of the pool per hour.

#### Step 3: Work done in the first 30 minutes
In the first 30 minutes (or 0.5 hours), the amount of the pool filled with all three pipes open is:
\[
\text{Amount filled} = 0.194 \times 0.5 = 0.097 \text{ (approximately)}.
\]
So, about 9.7% of the pool is filled in the first 30 minutes.

#### Step 4: Work done by pipes A and B after 3:30 PM
After 3:30 PM, only pipes A and B are open, and their combined rate is:
\[
\text{Rate of A and B} = \frac{1}{4} + \frac{1}{9} = \frac{9 + 4}{36} = \frac{13}{36} \text{ of the pool per hour}.
\]

#### Step 5: Remaining pool to fill
The remaining part of the pool to be filled is:
\[
1 - 0.097 = 0.903 \text{ (or 90.3% of the pool)}.
\]

The time required to fill the remaining part is:
\[
\text{Time} = \frac{0.903}{\frac{13}{36}} = 0.903 \times \frac{36}{13} \approx 2.5 \text{ hours}.
\]

#### Step 6: Time of completion
If you started the process at 3:00 PM, closed Pipe C at 3:30 PM, and then it took approximately 2.5 hours to fill the remaining pool, the pool would be fully filled at:
\[
3:30 \, \text{PM} + 2.5 \, \text{hours} = 6:00 \, \text{PM}.
\]

### Conclusion:
The pool would be completely filled at approximately **6:00 PM**.

---

Would you like further clarification, or have any specific questions? Here are a few related questions:

1. How do you calculate rates when multiple pipes are working together?
2. What if the pool had a different shape—would the filling time change?
3. How would a delay in closing Pipe C affect the outcome?
4. How can you calculate the total volume of water if the pool's size is known?
5. What if Pipe C had a faster rate of emptying?

**Tip:** Always check the rates of flow (filling or emptying) carefully when working with systems involving multiple inlets and outlets.

My hotel had a swimming pool, and I wanted to host a poolside party for my birthday. I wanted to fill the pool using three pipes. Two pipes (A and B) could fill the pool in 4 hours and 9 hours. Another pipe can empty the pool in 6 hours. By mistake I opened all three pipes at 3 pm and realised at 3.30 pm. So, I closed it. When would the pool be completely filled with water?

Learn how to calculate the time required to fill a swimming pool using two filling pipes and one emptying pipe. This problem involves rates of work, calculating net rates, and determining time based on partial work. Suitable for students in Grades 9-10, this detailed solution explains each step with key formulas and concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation